Relationship Between Chaos and Self-similarity

A phase-space is an N dimensional space whose every point fully characterizes the state of a system. The phase space of the water-wheel system can be characterized by three variables^3.2. If we plot every state of the system in time according to these three variables, we come up with a trajectory which characterizes the behavior of the system over time. Now, if we would be able to predict how this trajectory moves in the phase space, we would be able to predict the behavior of the system. If the trajectory was a line moving on a simple 2 dimensional surface and the system was linear, by having three samples of this trajectory we would be able to predict the behavior of the system. However, there are places in the phase space where the trajectory seems to trace a very thin volume, and the volume is created by infinite stretching and folding of a surface. The shape underlying this strange ``surface'' is the cantor set, which is self-similar. This implies that the trajectory is moving on a shape with infinite amount of detail, meaning that a different direction could be taken according to infinitesimal differences in initial condition. In a linear system, a small error in initial condition could only cause an error proportional to the original error. However in a system like this, an error (i.e., our inability to measure conditions with infinite precision) could cause completely different directions to be predicted for the trajectory. It is important to note that there is no noise introduced into the system, and this interesting behavior can be seen on the computer by trying to predict the trajectory of the system by using the three differential equations characterizing the system. By changing the integration interval and initial conditions, we obtain completely different results, while we are usually used to obtaining more accurate results when we integrate over smaller segments of time.